Problem: $ A = \left[\begin{array}{rrr}-2 & 3 & 3 \\ 0 & 0 & 1\end{array}\right]$ $ B = \left[\begin{array}{rr}-1 & -2 \\ 5 & 0 \\ 4 & 2\end{array}\right]$ What is $ A B$ ?
Solution: Because $ A$ has dimensions $(2\times3)$ and $ B$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A B = \left[\begin{array}{rrr}{-2} & {3} & {3} \\ {0} & {0} & {1}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{-2} \\ {5} & \color{#DF0030}{0} \\ {4} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-2}\cdot{-1}+{3}\cdot{5}+{3}\cdot{4} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-2}\cdot{-1}+{3}\cdot{5}+{3}\cdot{4} & ? \\ {0}\cdot{-1}+{0}\cdot{5}+{1}\cdot{4} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-2}\cdot{-1}+{3}\cdot{5}+{3}\cdot{4} & {-2}\cdot\color{#DF0030}{-2}+{3}\cdot\color{#DF0030}{0}+{3}\cdot\color{#DF0030}{2} \\ {0}\cdot{-1}+{0}\cdot{5}+{1}\cdot{4} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-2}\cdot{-1}+{3}\cdot{5}+{3}\cdot{4} & {-2}\cdot\color{#DF0030}{-2}+{3}\cdot\color{#DF0030}{0}+{3}\cdot\color{#DF0030}{2} \\ {0}\cdot{-1}+{0}\cdot{5}+{1}\cdot{4} & {0}\cdot\color{#DF0030}{-2}+{0}\cdot\color{#DF0030}{0}+{1}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}29 & 10 \\ 4 & 2\end{array}\right] $